80 problems

Let $\mathcal{L}$ be a nilpotent Lie lattice considered in CSV/19, let $K$ be a number field, and for each decomposition type $(\mathbf{e},\mathbf{f})$ let…

Let $F$ be the field under consideration, let $k$ be a positive integer, and let $x_1,\ldots,x_k$ be arguments for the motivic multiple polylogarithm…

A valid function is a function $\pi:\mathbb{R}\to\mathbb{R}_+$ satisfying the conditions for validity in the infinite group relaxation, and an extreme function is a valid function…

Let $f$ be a Siegel cusp eigenform of genus $n$ and weight $k$, and let $L(Sp(f),s)$ be its spinor $L$-function. Write $w=kn-n(n+1)/2$ and let $\Gamma_{n,k}(s)$ be the gamma factor…

Functional-equation conjecture. One has $m_p^R=m_{w-p}^R$ for all $p$, hence $m_{w/2}\equiv\dim V\pmod 2$, and: (1) $L(\iota R,s)$ extends to an entire function except for a single…

Let $\widehat G_{n-1}^n$ be the configuration space used to define the Grassmannian $n$-logarithm, and let $a_i,b_j$ be the maps on this space appearing in its functional equations…

Let $P(z)$ and $Q(z)$ be complex polynomials, and let $a,b\in\mathbb{C}$ with $a\neq b$ and $P(a)=P(b)$, $Q(a)=Q(b)$. For $i\geq 1$, define … The composition condition asserts that…

Both-types invariance conjecture. The only such pairs are the quasi-arithmetic transforms

Let $\Psi_0(z^i(\mathbf x))$ be a functional on a fixed spacelike surface $x^0=x^0(\mathbf x)$, and let equation (14) be the generalized Schrödinger equation governing its evolutio…

Let $\Gamma$ be the quantum functional satisfying the local functional equation, and let $\Gamma^{(0)}$ denote the classical action in equation. For a symmetric regularization such…

Let $x\in R^n$ and $t\in R^r$, let $\Omega\subseteq R^r$ be open, and consider distribution solutions of the functional equation (1.1) satisfying assumptions A-1 through A-5. Let…

Let $f\colon \mathbb{R}\to\mathbb{R}$ satisfy … where $\alpha$ is a given nonzero real number. Zero conjecture. Every solution $f$ of this equation has a zero. The conjecture conce…

Matkowski's conjecture. The function $f$ solves this equation if and only if there exist $a,b\in\mathbb{R}$ with $a\neq b$ such that

Let $G$ be a reductive group and let $V\in{\operatorname{Rep}}(G)$, with $V^\vee$ its dual representation. Let ${\mathcal{P}}_V^{\operatorname{norm}}$ denote the normalized period…

Let $X$ be a regular scheme, projective and flat over $\operatorname{Spec}(\mathbb{Z})$, and equidimensional of relative dimension $d$. Let $\zeta(X,s)$ be its Hasse--Weil zeta fun…

Halanay–Smith criterion. Let $M\in\mathcal G$. The following statements are equivalent:

Let $\zeta\coloneqq e^{\mathrm{i}2\pi/3}$, and consider functions $h$ that are analytic or holomorphic for $|z|<1$. Berg–Meinardus conjecture. The only solutions of the homogeneous…

Logarithmic-map discrepancy conjecture. The discrepancy is

Sine-map discrepancy conjecture. The discrepancy is

Vanishing-discrepancy conjecture. We conjecture that $\delta_1=0$ and that the identity $g_1(x)=\widetilde{g}_1(x)$ holds for every initial value $x=x_0$.

Let $A,B,I,J$ be the convex solutions of the Abel equations … … with the domains and normalizations described in the source: $A$ has a minimum at $x=1/2$, $B$ has a minimum at…

Matyas's conjecture. If $\varrho$ is a homogeneous polynomial of degree at least $4$ satisfying the displayed functional equation, then it satisfies the displayed identity. The que…

The paper considers the production of a function $f$ using a higher-order derivative, with the parameter $k$ specifying the case under consideration. Conjecture. When $k=2$, the or…

A functional identity of the dilogarithm is an equality involving dilogarithms whose arguments are rational functions. Wojtkowiak's result, as summarized in the source, states that…

Multiple polylogarithms satisfy functional equations, including those arising from the quadrangular polylogarithm functional equation. Functional-equation conjecture. All functiona…